Feedback and Robust Local Readout of a Fixed Four-State Quantum Source

This paper develops local quantum-reader decision problems within Shadow Theory’s source–readout programme. It asks when feedback improves access to a target distinction in a fixed quantum source, and how reliably that distinction can be read under detector noise and calibration uncertainty. The source supplies independent copies conditional on one hidden label, held fixed throughout the experiment, from four equatorial qubit preparations. The task is to identify their common positive or negative sign, rather than reconstruct the full preparation label. Although all four preparations have identical separate-mode intensities, phase-sensitive local measurements provide access to the target. The analysis compares adaptive readers, whose measurement settings depend on earlier records, with randomized nonadaptive readers, whose complete schedules are selected before outcomes are observed. Each measurement consumes one fresh copy. Every protocol must return a sign at a fixed deadline, retaining all settings, outcomes, nulls and specified timing records. Neither class uses collective measurements or coherent quantum memory between specimens. The main results establish equal optimal worst-case error for both reader classes through three copies across the shared efficiency and visibility model. At four copies, explicit rational feedback policies outperform every randomized nonadaptive schedule over the full continuous equatorial measurement menu, in both the ideal model and a specified noisy model. The comparison is supported by continuous-menu lower certificates rather than measurement-grid searches alone. For a one-percent worst-case error target, the minimum copy count is narrowed to either 13 or 14 for each reader class in the nominal model, and either 16 or 17 under the specified asymmetric gain and phase uncertainty. These adjacent thresholds remain unresolved. Further results give an exact robust repeated-angle frontier, extend the comparison to randomized choices of a fixed repeated angle, and identify convex-source conditions under which one reader is globally sufficient. The accompanying technical supplement and reproducibility materials provide measurement-allocation benchmarks, expanded finite-menu comparisons, compact feedback policies, exact optimization certificates, and timing and calibration interfaces. The results quantify target access under explicitly stated preparation, measurement and record-access conditions. They use standard quantum probabilities within Shadow Theory’s research programme and distinguish copy savings from total apparatus, control or calibration savings.